for moving emblems from one place to another until certain kinds of results
are obtained. Platonic ideologies to the contrary, mathematics does not exist
purely in the mind; it is a set of practices, developed by generations of tinkering,
and an integral part of these practices is the physical “equipment” with which
they are connected. This is not so distant from our implicit definition of a
machine as a material entity, for every machine consists in the combination of
the physical object with skills for manipulating it. Sets of mathematical symbols
on paper, lined up in equations and rearranged according to rules, represent a
practical activity rather than simply a set of abstract ideas.
Turning mathematics into a problem-solving machine was not simply a
matter of new notation, although the emergence of symbolism did take place
at the time of the mathematical revolution. There had been episodes of synco-
pated, or abbreviated, algebra before (Diophantus ca. 250 c.e., Brahmagupta
ca. 630 c.e.), but these had not been consistently followed up, and most
of Chinese, Greek, and Muslim mathematics was argued out in words, with
the assistance of geometrical diagrams. In medieval Christendom, Fibonacci’s
math (ca. 1200) was rhetorical; so were the difficult and involved proofs of
Swineshead “the Calculator,” as well as the generalizing efforts of Regiomon-
tanus in the mid-1400s. Syncopated forms arose in the early 1500s in arith-
metic and algebra, especially with the “reckoning masters” of the commercial
German cities, and the symbolic apparatus moved rapidly forward with Viète,
reaching what became more or less the standard modern form with Descartes.
There are several reasons why we should not take notation per se as the
key to the mathematical takeoff. Much of the development of notation took
place not among mathematicians producing creative new results, but in the
textbooks explaining commercial arithmetic which proliferated from the 1480s
onward.^13 Still less should we regard the spread of Hindu-Arabic numerals,
with place notation and the zero sign, as the key. These were not associated
with higher mathematics in their place of origin; they provoked no creativity
at all when they became available in Byzantium, and they were known in
medieval Europe centuries before the mathematical takeoff of the 1500s (Kazh-
dan and Epstein, 1985: 145; Smith, 1951). On the side of the intellectuals, the
mathematical “machinery” which began to automate the solution of equations
was often formulated without benefit of the more concise notation. Cardan’s
exposition was rhetorical, but he gave general rules for solving equations by
manipulating and substituting terms so as to turn unknown expressions into
the form of solvable ones. Viète was more syncopated than symbolic, still using
some verbal argument; but he clearly recognized the generality of the un-
knowns, distinguishing assumed unknowns from assumed givens. Even with
this unwieldy apparatus, he developed the machinery of problem-solving pro-
cedure by creating new equations to substitute into old ones. Pascal, as late as
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